Math Table

Nontrivial Holomorphic Structures on the Quantum Sphere

Speaker: Lillian MacArthur

Abstract: The area of Noncommutative Geometry is quite new, and even newer is the concept of a complex structure in Noncommutative Geometry. In previous literature, various complex structures have been defined within the noncommutative framework, with various results found, including the Quantum Sphere. However, all previous investigations into the structure of the Quantum Sphere have always found its algebraic and analytic properties to be not too dissimilar from the classical Riemann Sphere. Here, we demonstrate that in regard to holomorphic structures that can be defined on the space, the Quantum Sphere is indeed very different from the Riemann Sphere.

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Bézout's theorem and the birth of algebraic topology

Speaker: Joe Harris

Abstract: Bézout's theorem is a direct generalization of the Fundamental Theorem of Algebra to polynomials in multiple variables: it predicts the number of solutions of a system of n polynomials in n variables. It was originally proved by algebraic methods, using resultants to reduce the problem to the FTA; but Poincaré saw in it the germ of an idea that he would ultimately develop into what we now recognize as homology, cohomology, cup product and Poincaré duality. In this talk, I'll try to sketch how this came about, and why Bézout's theorem is ultimately an elementary theorem in algebraic topology.